Parabolic frequency on Ricci flows

TL;DR

This paper introduces a parabolic frequency for heat equation solutions on Ricci flows, proves its monotonicity, and uses it to establish backwards uniqueness, offering new tools for analyzing parabolic PDEs in geometric flows.

Contribution

It defines a new parabolic frequency on Ricci flows, proves its monotonicity, and applies it to demonstrate backwards uniqueness for solutions of parabolic equations.

Findings

Frequency is monotonic along Ricci flows.

Monotonicity leads to a simple proof of backwards uniqueness.

Bounds on frequency derivatives extend backwards uniqueness to general parabolic equations.

Abstract

This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.